Title: Siegel's Lemma for polynomials irreducible over Q
Speaker: Michael Filaseta
Abstract: Siegel's Lemma is a statement about the finiteness of the number of integer points on an irreducible curve f(x,y) = 0 when the genus of the curve is at least 1. Here, we want f(x,y) to have rational coefficients and for irreducibility to be over the complex numbers (in C[x,y]). In this talk, we discuss how to restrict to the case where instead f(x,y) is irreducible over the rationals. The resulting statement is different but will give us what we want in forthcoming talks on deducing Hilbert's Irreducibility Theorem from Siegel's Lemma. The idea is to make this connection without much in the way of machinery, so the talk is intended for graduate students and faculty alike.
Title: Universal Hilbert Sets
Speaker: Michael Filaseta
Abstract: A universal Hilbert set is an infinite set S of integers with the property that for every f(x,y) with integer coefficients with f(x,y) irreducible in Q[x,y] and of degree at least 1 in x, we have that for all but finitely many t in S, the polynomial f(x,t) is irreducible in Q[x]. We describe a set S with this property that has asymptotic density 1 in the integers. Then we show an important connection that S has with Siegel's Lemma.
Title: Universal Hilbert Sets, Part II
Speaker: Michael Filaseta
Abstract: After reviewing a little from last week, we finish a proof of a lemma associated with Hilbert's Irreducibility Theorem and Siegel's Lemma. We also prove an additional result that encapsulates the use of Siegel's Lemma in proving Hilbert's Irreducibility Theorem. After this result, there will be no need to recall the statement of Siegel's Lemma or the definition of our example of a universal Hilbert set, as this result indirectly embeds all the information we will need.
Title: What is the height of two points in the plane?
Speaker: Frank Thorne
Abstract: A classical question in arithmetic geometry is to count the number of rational points of bounded height on algebraic varieties.
After reviewing some classical cases, I'll discuss this problem for the Hilbert scheme of two points in the plane. This quickly turns into a lattice point counting problem, and invites related questions as well as arithmetic applications.
I'll spend a lot of the talk explaining what all the words mean.
Joint work with Jesse Kass.
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Speaker: Lola Thompson (Oberlin College)
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